More Like this

1. Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

   https://doi.org/10.3934/krm.2022012  

   Nazarov, Fedor; Zumbrun, Kevin (January 2022, Kinetic and Related Models)

   We establish an instantaneous smoothing property for decaying solutions on the half-line \begin{document}$$ (0, +\infty) $$\end{document} of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \begin{document}$ H^1 $$\end{document} stable manifolds of such equations, showing that \begin{document}$$ L^2_{loc} $$\end{document} solutions that remain sufficiently small in \begin{document}$$ L^\infty $$\end{document} (i) decay exponentially, and (ii) are \begin{document}$$ C^\infty $$\end{document} for \begin{document}$$ t>0 $$\end{document}, hence lie eventually in the \begin{document}$$ H^1 $$\end{document}$ stable manifold constructed by Pogan and Zumbrun. 

   more » « less
2. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

   https://doi.org/10.3934/dcdss.2022016  

   Wu, Yixuan; Zhang, Yanzhi (January 2022, Discrete & Continuous Dynamical Systems - S)

   In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$$ (- \Delta)^\frac{{ \alpha}}{{2}} $$\end{document} for \begin{document}$$ \alpha \in (0, 2) $$\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}$$ {\mathcal O}(h^2) $$\end{document}, while \begin{document}$$ {\mathcal O}(h^4) $$\end{document} for quadratic basis functions with \begin{document}$ h $$\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$$ \alpha \in (0, 2) $$\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$$ u \in C^{m, l}(\bar{ \Omega}) $$\end{document} for \begin{document}$$ m \in {\mathbb N} $$\end{document} and \begin{document}$$ 0 < l < 1 $$\end{document}, our method has an accuracy of \begin{document}$$ {\mathcal O}(h^{\min\{m+l, \, 2\}}) $$\end{document} for constant and linear basis functions, while \begin{document}$$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $$\end{document}$ for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency. 

   more » « less
3. From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae

   https://doi.org/10.3934/eect.2022007  

   Triggiani, Roberto; Wan, Xiang (January 2022, Evolution Equations and Control Theory)

   We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$ g $$\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $$\end{document}, which, moreover, in the canonical case \begin{document}$$ \gamma = 0 $$\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$$ \gamma = 0 $$\end{document} or \begin{document}$$ 0 \neq \gamma \in L^{\infty}(\Omega) $$\end{document}, since \begin{document}$$ \gamma \neq 0 $$\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$$ g $$\end{document} "smoother" than \begin{document}$$ L^2(\Sigma) $$\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$$ L^2(0, T;L^2(\Gamma)) $$\end{document}, and [44] for control less regular in space than \begin{document}$$ L^2(\Gamma) $$\end{document}$. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2]. 

   more » « less
4. Thermodynamic formalism for dispersing billiards

   https://doi.org/10.3934/jmd.2022013  

   Baladi, Viviane; Demers, Mark F. (January 2022, Journal of Modern Dynamics)

   For any finite horizon Sinai billiard map \begin{document}$ T $$\end{document} on the two-torus, we find \begin{document}$$ t_*>1 $$\end{document} such that for each \begin{document}$$ t\in (0,t_*) $$\end{document} there exists a unique equilibrium state \begin{document}$$ \mu_t $$\end{document} for \begin{document}$$ - t\log J^uT $$\end{document}, and \begin{document}$$ \mu_t $$\end{document} is \begin{document}$$ T $$\end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for \begin{document}$$ - \log J^uT $$\end{document}.) We show that \begin{document}$$ \mu_t $$\end{document} is exponentially mixing for Hölder observables, and the pressure function \begin{document}$$ P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\} $$\end{document} is analytic on \begin{document}$$ (0,t_*) $$\end{document}. In addition, \begin{document}$$ P(t) $$\end{document} is strictly convex if and only if \begin{document}$$ \log J^uT $$\end{document} is not \begin{document}$$ \mu_t $$\end{document}-a.e. cohomologous to a constant, while, if there exist \begin{document}$$ t_a\ne t_b $$\end{document} with \begin{document}$$ \mu_{t_a} = \mu_{t_b} $$\end{document}, then \begin{document}$$ P(t) $$\end{document} is affine on \begin{document}$$ (0,t_*) $$\end{document}. An additional sparse recurrence condition gives \begin{document}$$ \lim_{t\downarrow 0} P(t) = P(0) $$\end{document}$. 

   more » « less
5. A measure model for the spread of viral infections with mutations

   https://doi.org/10.3934/nhm.2022015  

   Gong, Xiaoqian; Piccoli, Benedetto (January 2022, Networks and Heterogeneous Media)

   Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible \begin{document}$ S $$\end{document} and removed \begin{document}$$ R $$\end{document} populations by ODEs and the infected \begin{document}$$ I $$\end{document} population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for \begin{document}$$ S $$\end{document} and \begin{document}$$ R $$\end{document} contains terms that are related to the measure \begin{document}$$ I $$\end{document}$. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases. 

   more » « less